Subgroups — C₂×#11720297 #13139462

Order 1728.

562 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
4	#11720297	C₂	1	4	yes	maximal, direct product factor
5	C₂×#63565	—	3	1	no	maximal
6	C₂×#63395	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₉×Dic₃	—	2	1	no	—
9	C₂²×C₃×Dic₉	—	2	1	no	—
10	C₂×C₉×Dic₆	—	2	1	no	—
11	C₂×C₉×Dic₆	—	2	2	no	—
12	C₂×C₉×Dic₆	C₄	1	4	yes	complemented
13	C₂×C₉×Dic₆	C₂²	1	2	yes	—
14	C₂×C₃⋊(C₉⋊C₈)	—	2	2	no	—
15	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
16	C₃×C₄⋊Dic₉	—	2	1	no	—
17	C₃×C₄⋊Dic₉	C₂²	1	2	yes	—
18	C₂²×C₃×Dic₆	D₃	1	1	yes	characteristic
19	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
20	C₂×C₃×C₄⋊Dic₃	—	3	1	no	—
21	C₂²×Q₈×C₉	—	3	1	no	—
22	#63565	—	3	4	no	—
23	C₂×C₄⋊Dic₉	D₃	1	1	yes	characteristic
24	C₂²×C₉⋊C₈	—	3	1	no	—
25	C₂²×C₉⋊C₈	—	6	1	no	—
26	#63395	—	3	4	no	—
27	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₂×C₉×Dic₃	—	2	4	no	—
33	C₂×C₉×Dic₃	—	2	2	no	—
34	C₂×C₃×Dic₉	—	2	2	no	—
35	C₂×C₃×Dic₉	—	4	2	no	—
36	C₉×Dic₆	—	2	4	no	—
37	C₉×Dic₆	—	2	2	no	—
38	C₉×Dic₆	C₂×C₄	1	4	yes	complemented
39	C₃⋊(C₉⋊C₈)	—	2	2	no	—
40	C₂×#835	—	9	1	no	—
41	C₂×#810	—	9	1	no	—
42	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
43	C₂²×C₃×Dic₃	—	2	1	no	—
44	C₂²×C₃×Dic₃	—	6	1	no	—
45	C₂×C₃×Dic₆	—	2	1	no	—
46	C₂×C₃×Dic₆	—	2	2	no	—
47	C₂×C₃×Dic₆	Dic₃	1	4	yes	—
48	C₂×C₃×Dic₆	C₂×D₃	1	2	yes	—
49	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
50	C₂×C₃⋊(C₃⋊C₈)	—	6	2	no	—
51	C₃×C₄⋊Dic₃	—	3	2	no	—
52	C₃×C₄⋊Dic₃	—	6	1	no	—
53	C₂×Q₈×C₉	—	3	4	no	—
54	C₂×Q₈×C₉	—	3	2	no	—
55	C₂×Q₈×C₉	—	6	2	no	—
56	C₂×Q₈×C₉	—	6	1	no	—
57	C₂²×C₄×C₉	—	2	1	no	—
58	C₂²×C₄×C₉	—	6	1	no	—
59	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
60	C₂²×Dic₉	—	2	1	no	—
61	C₄⋊Dic₉	—	2	1	no	—
62	C₄⋊Dic₉	C₂×D₃	1	2	yes	—
63	C₂×C₉⋊C₈	—	3	2	no	—
64	C₂×C₉⋊C₈	—	6	2	no	—
65	C₂×C₉⋊C₈	—	6	4	no	—
66	C₂×C₉⋊C₈	—	6	2	no	—
67	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₃×C₉	SD₁₆	1	2	yes	—
69	C₂²×C₃×C₉	Q₁₆	1	2	yes	—
70	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
71	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
72	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
73	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
74	C₉×Dic₃	—	2	4	no	—
75	C₃×Dic₉	—	4	2	no	—
76	C₂²×C₃×Q₈	—	3	1	no	—
77	C₂²×Dic₆	D₉	1	1	yes	characteristic
78	C₂×C₃×C₄⋊C₄	—	9	1	no	—
79	C₂×C₄⋊Dic₃	—	3	1	no	—
80	C₂²×C₃⋊C₈	—	9	1	no	—
81	C₂²×C₃⋊C₈	—	9	1	no	—
82	C₂²×C₃⋊C₈	—	18	1	no	—
83	#835	—	9	4	no	—
84	#810	—	9	4	no	—
85	C₂³×C₃²	#67	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
87	C₂×C₃²×C₄	#67	1	1	yes	characteristic
88	C₂×C₃²×C₄	#67	1	2	yes	—
89	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
90	C₂×C₃×Dic₃	—	2	4	no	—
91	C₂×C₃×Dic₃	—	2	2	no	—
92	C₂×C₃×Dic₃	—	6	2	no	—
93	C₂×C₃×Dic₃	—	12	2	no	—
94	C₃×Dic₆	—	2	4	no	—
95	C₃×Dic₆	—	2	2	no	—
96	C₃×Dic₆	C₂×Dic₃	1	4	yes	—
97	C₂³×C₉	—	2	1	no	—
98	C₂³×C₉	#67	1	1	yes	characteristic
99	C₃⋊(C₃⋊C₈)	—	6	2	no	—
100	Q₈×C₉	—	3	4	no	—
101	Q₈×C₉	—	6	4	no	—
102	Q₈×C₉	—	6	2	no	—
103	C₂×C₄×C₉	—	2	2	no	—
104	C₂×C₄×C₉	—	2	1	no	—
105	C₂×C₄×C₉	—	2	2	no	—
106	C₂×C₄×C₉	—	2	1	no	—
107	C₂×C₄×C₉	—	6	4	no	—
108	C₂×C₄×C₉	—	6	2	no	—
109	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
110	C₂×C₄×C₉	D₁₂	1	2	yes	—
111	C₂×C₄×C₉	#67	1	1	yes	characteristic
112	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
113	C₂×Dic₉	—	2	2	no	—
114	C₂×Dic₉	—	4	2	no	—
115	C₉⋊C₈	—	6	4	no	—
116	C₉⋊C₈	—	6	2	no	—
117	C₂×#103	—	27	1	no	Sylow
118	C₂×C₃×C₉	#103	1	4	yes	complemented
119	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
122	C₂×C₃×Q₈	—	3	4	no	—
123	C₂×C₃×Q₈	—	3	2	no	—
124	C₂×C₃×Q₈	—	6	1	no	—
125	C₂×C₃×Q₈	—	6	2	no	—
126	C₂²×C₃×C₄	—	2	1	no	—
127	C₂²×C₃×C₄	—	6	1	no	—
128	C₂²×C₃×C₄	—	18	1	no	—
129	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
130	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
131	C₂²×Dic₃	—	2	1	no	—
132	C₂²×Dic₃	—	6	1	no	—
133	C₂×Dic₆	—	2	1	no	—
134	C₂×Dic₆	—	2	2	no	—
135	C₂×Dic₆	Dic₉	1	4	yes	complemented
136	C₂×Dic₆	C₂×D₉	1	2	yes	—
137	C₃×C₄⋊C₄	—	9	2	no	—
138	C₃×C₄⋊C₄	—	18	1	no	—
139	C₄⋊Dic₃	—	3	2	no	—
140	C₄⋊Dic₃	—	6	1	no	—
141	C₂×C₃⋊C₈	—	9	2	no	—
142	C₂×C₃⋊C₈	—	9	2	no	—
143	C₂×C₃⋊C₈	—	18	2	no	—
144	C₂×C₃⋊C₈	—	18	4	no	—
145	C₂×C₃⋊C₈	—	18	2	no	—
146	C₂×C₃⋊C₈	—	18	2	no	—
147	C₂²×C₃²	#215	1	2	yes	—
148	C₂²×C₃²	#216	1	2	yes	—
149	C₂²×C₃²	#217	1	2	yes	—
150	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
151	C₃²×C₄	#217	1	2	yes	—
152	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
153	C₃²×C₄	C₂×#67	1	1	yes	characteristic
154	C₃×Dic₃	—	2	4	no	—
155	C₃×Dic₃	—	12	2	no	—
156	C₂²×C₉	—	2	2	no	—
157	C₂²×C₉	—	2	2	no	—
158	C₂²×C₉	—	2	2	no	—
159	C₂²×C₉	—	2	1	no	—
160	C₂²×C₉	(C₂×D₃)⋊C₄	1	2	yes	—
161	C₂²×C₉	#214	1	2	yes	—
162	C₂²×C₉	#216	1	2	yes	—
163	C₂²×C₉	C₂×#67	1	1	yes	characteristic
164	C₄×C₉	—	2	2	no	—
165	C₄×C₉	—	2	1	no	—
166	C₄×C₉	—	2	1	no	—
167	C₄×C₉	—	6	4	no	—
168	C₄×C₉	(C₂×D₃)⋊C₄	1	2	yes	—
169	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
170	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
171	Dic₉	—	4	2	no	—
172	C₂²×Q₈	—	3	1	no	—
173	C₂²×C₈	—	27	1	no	—
174	C₂×C₄⋊C₄	—	9	1	no	—
175	#103	—	27	4	no	—
176	C₃×C₉	C₂×#103	1	1	yes	characteristic, complemented, Sylow
177	C₂³×C₃	—	2	1	no	—
178	C₂³×C₃	#611	1	1	yes	characteristic
179	C₂³×C₃	#625	1	1	yes	characteristic
180	C₃×Q₈	—	3	4	no	—
181	C₃×Q₈	—	6	4	no	—
182	C₃×Q₈	—	6	2	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	6	4	no	—
188	C₂×C₃×C₄	—	6	2	no	—
189	C₂×C₃×C₄	—	18	2	no	—
190	C₂×C₃×C₄	—	36	2	no	—
191	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
192	C₂×C₃×C₄	#611	1	1	yes	characteristic
193	C₂×C₃×C₄	#611	1	2	yes	—
194	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
195	C₂×C₃×C₄	D₃×Dic₃	1	2	yes	—
196	C₂×C₃×C₄	#625	1	1	yes	characteristic
197	C₂×C₃×C₄	#626	1	2	yes	—
198	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
199	C₂×Dic₃	—	2	4	no	—
200	C₂×Dic₃	—	2	2	no	—
201	C₂×Dic₃	—	6	2	no	—
202	C₂×Dic₃	—	12	2	no	—
203	Dic₆	—	2	4	no	—
204	Dic₆	—	2	2	no	—
205	Dic₆	C₂×Dic₉	1	4	yes	complemented
206	C₃⋊C₈	—	18	4	no	—
207	C₃⋊C₈	—	18	2	no	—
208	C₃⋊C₈	—	18	2	no	—
209	C₂×C₃²	#835	1	4	yes	—
210	C₂×C₃²	C₂×#215	1	1	yes	characteristic
211	C₂×C₃²	C₂×#216	1	1	yes	characteristic
212	C₂×C₃²	C₂×#217	1	1	yes	characteristic
213	C₂×C₉	—	2	4	no	—
214	C₂×C₉	—	2	1	no	—
215	C₂×C₉	—	2	1	no	—
216	C₂×C₉	—	2	1	no	—
217	C₂×C₉	#810	1	4	yes	complemented
218	C₂×C₉	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
219	C₂×C₉	C₂×#214	1	1	yes	characteristic
220	C₂×C₉	C₂×#216	1	1	yes	characteristic
221	C₂×Q₈	—	3	4	no	—
222	C₂×Q₈	—	3	2	no	—
223	C₂×Q₈	—	6	1	no	—
224	C₂×Q₈	—	6	2	no	—
225	C₂²×C₄	—	6	1	no	—
226	C₂²×C₄	—	18	1	no	—
227	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
228	C₂×C₈	—	27	2	no	—
229	C₂×C₈	—	54	2	no	—
230	C₄⋊C₄	—	9	2	no	—
231	C₄⋊C₄	—	18	1	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	—	2	2	no	—
236	C₂²×C₃	#3674	1	2	yes	—
237	C₂²×C₃	#3675	1	2	yes	—
238	C₂²×C₃	#3676	1	2	yes	—
239	C₂²×C₃	C₂×#611	1	1	yes	characteristic
240	C₂²×C₃	#3715	1	2	yes	—
241	C₂²×C₃	#3718	1	2	yes	—
242	C₂²×C₃	(C₂×D₃)⋊Dic₃	1	2	yes	—
243	C₂²×C₃	C₂×#625	1	1	yes	characteristic
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	—	6	4	no	—
248	C₃×C₄	—	36	2	no	—
249	C₃×C₄	#3676	1	2	yes	—
250	C₃×C₄	C₂²×Dic₉	1	1	yes	characteristic
251	C₃×C₄	C₂×#611	1	1	yes	characteristic
252	C₃×C₄	(C₂×D₃)⋊Dic₃	1	2	yes	—
253	C₃×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
254	C₃×C₄	C₂×#626	1	1	yes	characteristic
255	Dic₃	—	2	4	no	—
256	Dic₃	—	12	2	no	—
257	C₃²	C₂×#835	1	1	yes	characteristic
258	C₉	—	2	1	no	—
259	C₉	C₂×#810	1	1	yes	characteristic, complemented
260	C₂³	#6194	1	1	yes	characteristic
261	Q₈	—	3	4	no	—
262	Q₈	—	6	2	no	—
263	Q₈	—	6	4	no	—
264	C₂×C₄	—	6	2	no	—
265	C₂×C₄	—	6	4	no	—
266	C₂×C₄	—	18	2	no	—
267	C₂×C₄	—	36	2	no	—
268	C₂×C₄	D₃×Dic₉	1	2	yes	—
269	C₂×C₄	#6194	1	1	yes	characteristic
270	C₂×C₄	#6195	1	2	yes	—
271	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
272	C₈	—	54	2	no	—
273	C₂×C₃	—	2	1	no	—
274	C₂×C₃	—	2	4	no	—
275	C₂×C₃	—	2	1	no	—
276	C₂×C₃	—	2	1	no	—
277	C₂×C₃	#63395	1	4	yes	complemented
278	C₂×C₃	C₂×#3674	1	1	yes	characteristic
279	C₂×C₃	C₂×#3675	1	1	yes	characteristic
280	C₂×C₃	C₂×#3676	1	1	yes	characteristic
281	C₂×C₃	#63565	1	4	yes	—
282	C₂×C₃	C₂×#3715	1	1	yes	characteristic
283	C₂×C₃	C₂×#3718	1	1	yes	characteristic
284	C₂×C₃	C₂×(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
285	C₂²	#88425	1	2	yes	—
286	C₂²	#88427	1	2	yes	—
287	C₂²	(C₂×D₃)⋊Dic₉	1	2	yes	—
288	C₂²	C₂×#6194	1	1	yes	characteristic
289	C₄	—	6	4	no	—
290	C₄	—	36	2	no	—
291	C₄	(C₂×D₃)⋊Dic₉	1	2	yes	—
292	C₄	C₂×D₃×Dic₉	1	1	yes	characteristic
293	C₄	C₂×#6195	1	1	yes	characteristic
294	C₃	—	2	1	no	—
295	C₃	C₂×#63395	1	1	yes	characteristic, complemented
296	C₃	C₂×#63565	1	1	yes	characteristic
297	C₂	#11720297	1	4	yes	direct product factor
298	C₂	C₂×#88425	1	1	yes	characteristic
299	C₂	C₂×#88427	1	1	yes	characteristic
300	C₂	C₂×(C₂×D₃)⋊Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6194
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle, prime radical	C₂³×C₃²	#67
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	C₂×#103
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. #11720297	C₂	direct (N × H)
12. C₂×C₉×Dic₆	C₄	semidirect
38. C₉×Dic₆	C₂×C₄	semidirect
118. C₂×C₃×C₉	#103	semidirect
135. C₂×Dic₆	Dic₉	semidirect
176. C₃×C₉	C₂×#103	semidirect
205. Dic₆	C₂×Dic₉	semidirect
217. C₂×C₉	#810	semidirect
259. C₉	C₂×#810	semidirect
277. C₂×C₃	#63395	semidirect
295. C₃	C₂×#63395	semidirect
297. C₂	#11720297	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₉×Dic₆	2
C₂²×C₃⋊(C₉⋊C₈)	2
C₂×C₃×C₄⋊Dic₉	2
#11720297	2
C₂×#63565	3
C₂×#63395	3

Subgroups — C2×#11720297 #13139462

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720297 #13139462